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Back-ordered inventory model with inflation in a cloudy-fuzzy environment

  • * Corresponding Author: Sankar Kumar Roy

    * Corresponding Author: Sankar Kumar Roy 

The author, Haripriya Barman is very much thankful to University Grants Commission (UGC) of India for providing financial support to continue this research work under Junior Research Fellowship scheme with UGC-reference number: [UGC-Ref.No.: 1166/(CSIR-UGC NET DEC.2017)] dated 08/02/2019

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  • In this paper, an Economic Production Quantity model for deteriorating items with time-dependent demand and shortages including partially back-ordered is developed under a cloudy-fuzzy environment. At first, we develop a crisp model by considering linearly time-dependent demand with constant deterioration rate, constant inflation rate and shortages under partially back-ordered, then we fuzzify the model to archive a decision under the cloudy-fuzzy (extension of fuzziness) demand rate, inflation rate, deterioration rate and the partially back-ordered rate which are followed by their practical applications. In this model, we assume ambiances where cloudy normalized triangular fuzzy number is used to handle the uncertainty in information which is coming from the data. The main purpose of our study is to defuzzify the total inventory cost by applying Ranking Index method of fuzzy numbers as well as cloudy-fuzzy numbers and minimize the total inventory cost of crisp, fuzzy, and cloudy-fuzzy model. Finally, a comparative analysis among crisp, fuzzy and cloudy-fuzzy total cost is carried out in this paper. Numerical example, sensitivity analysis, and managerial insights are elaborated to justify the usefulness of the new approach. A comparative inquiry of the numerical result with a new existing paper is also carried out. This paper ends with a conclusion along with advantages and limitations of our solution approach, and an outlook towards possible future studies.

    Mathematics Subject Classification: Primary: 90B05; Secondary: 90C70, 90C25.

    Citation:

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  • Figure 1.  Graphical representation of our inventory model

    Figure 2.  Graphical representation of the convexity total cost for crisp model. The figure represents the total cost $ Z(T, T_1, T_2) $, $ T_1 $ and $ T_2 $, along the axis of blue colour, the axis of red colour and the axis of green colour, respectively

    Figure 3.  Graphical representation of the convexity of total cost of our model in fuzzy environment. The figure represents the total cost $ I(\tilde{Z}) $, $ T_1 $ and $ T_2 $, along the axis of blue colour, the axis of red colour and the axis of green colour, respectively

    Figure 4.  Graphical representation of the convexity of total cost of our model in cloudy-fuzzy environment. The figure represents the total cost $ CI(\tilde{Z}) $, $ T_1 $ and $ T_2 $, along the axis of blue colour, the axis of red colour and the axis of green colour, respectively

    Figure 5.  Cost variation with respect to cycle-time variation for cloudy-fuzzy model

    Figure 6.  Comparison of cloudy solution with crisp and fuzzy solution

    Figure 7.  Graph of Set up cost vs inventory total cost

    Figure 8.  Graph of Inflation rate vs inventory total cost

    Table 1.  Contribution of various authors related to inventory models

    Author(s)Variable demand RateFuzzy demand RateCloudy Fuzzy Demand Rate Fuzzy DeteriorationCloudy Fuzzy DeteriorationFuzzy Inflationcloudy Fuzzy InflationVariable Holding costFuzzy Back order Ratecloudy Back order Rate
    De et al. (2003)$ \surd $$ \surd $
    De and Goswami (2006)$ \surd $$ \surd $
    Jaggi et al. (2012)$ \surd $$ \surd $
    Dutta and Kumar (2013)$ \surd $$ \surd $
    Nadjfi (2013)$ \surd $
    Shaboni et al. (2014)$ \surd $
    Pervin et al. (2015)$ \surd $
    Kumar and Rajput (2015)$ \surd $$ \surd $
    Pervin et al. (2016)$ \surd $$ \surd $
    De and Mahata (2017)$ \surd $ $ \surd $ $ \surd $
    Parvin et al. (2017)$ \surd $$ \surd $
    Kumar and Kumar (2017)$ \surd $$ \surd $$ \surd $
    Pervin et al. (2018)$ \surd $$ \surd $
    Our paper$ \surd $$ \surd $$ \surd $$ \surd $$ \surd $$ \surd $$ \surd $$ \surd $$ \surd $$ \surd $
     | Show Table
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    Table 2.  Optimal solutions of our models

    Model$T^*$(week)${T_1}^*$(week)${T_2}^*$(week)Minimum cost $ Z^*($) $$ d_f = \frac{u_b-l_b}{2M}$$ CI=\frac{\mbox{ln}(1+T)}{T} $
    Crisp4.219990.01019963.16691206.931
    Fuzzy4.841940.009158842.80802213.6340.16
    Cloudy fuzzy5.724180.01013973.15275169.7520.14
     | Show Table
    DownLoad: CSV

    Table 3.  Cycle-time variation in crisp model and fuzzy model

    Cycle timeCrisp modelFuzzy model
    $T_1$$T_2$$Z^*$$T_1$$T_2$$Z^*$
    20.01028813.18061253.5430.009213172.82304265.894
    30.01024833.17444218.9740.009194082.81776228.391
    40.01020843.16827207.8670.009174962.81248215.966
    $\textbf{5}$$\textbf{0.0101685}$$\textbf{3.16208}$$\textbf{206.271}$$\textbf{0.00915582}$$\textbf{2.80719}$$\textbf{213.7}$
    60.01012863.15589209.5360.009136652.80188216.623
    70.01008873.14968215.6690.009117462.79657222.603
    80.01004883.14347223.6740.009098252.79125230.574
    90.01000893.13725232.9970.009079012.78591239.944
    100.009973.13101243.3070.009059752.78057250.359
    * Bold represents optimal solution.
     | Show Table
    DownLoad: CSV

    Table 4.  Cycle-time variation in cloudy-fuzzy model

    Cycle time$ T_1 $$ T_2 $$ Z^* $
    20.01028813.18061233.896
    30.01018243.15873195.051
    40.01016613.1565177.841
    50.01015013.15431170.946
    60.01013573.15216169.901
    70.01012123.15005172.574
    80.0101073.14797177.808
    90.01009323.14592184.919
    100.01007963.1439193.475
    * Bold represents optimal solution.
     | Show Table
    DownLoad: CSV

    Table 5.  Sensitivity Analysis of the cloudy-fuzzy model

    Parameter$ \% $ changeNew value of parameter$ T $$ T_1 $$ T_2 $$ Z_* $$ \frac{Z_*-Z^*}{Z^*}\times100\% $
    $ B $+507508.107370.0101553.14775246.25945
    +306507.245030.0101173.14954217.98828.3
    +155756.526510.0101283.15104194.94214.74
    -154254.802460.01015363.15474141.517-0.17
    -303503.689460.01017113.15718108.416-36.19
    -502501.455170.01021023.1622345.2895-73.34
    $ c_1 $+5028.55.750630.01014583.15369169.34-0.33
    +3024.75.739970.01014343.15332169.505-0.23
    +1521.855.732040.01014163.15304169.628-0.16
    -3013.35.708620.01013593.15216169.9970.06
    -509.55.698370.01013333.15175170.1610.15
    $ \gamma $+5070.2555.727070.01017553.1383170.1950.17
    +300.2215.725930.01016113.14403170.0180.07
    +150.19555.725070.01015043.14837169.9060.03
    -150.14455.723260.01012923.15716169.617-0.17
    -300.1195.722310.01011873.16161169.302-0.4
    -500.0855.7210.0101053.1676169.302-0.4
    $ \rho $+500.2255.721390.0101093.16583169.355-0.32
    +300.1955.722540.01012123.16056169.514-0.23
    +150.17255.723370.01013043.15664169.633-0.16
    -150.12755.724960.01014913.14888169.87-0.02
    -300.1055.725730.01015863.14505169.9870.05
    -500.0755.726720.01017133.13998170.1430.14
    $ \alpha $+501.054.414410.01191043.11271191.32112.61
    +300.914.855470.01120163.12682183.8088.19
    +150.8055.251370.01067053.13897177.2474.32
    -150.5956.304280.00960943.16852161.137-5.16
    -300.497.04230.0090793.18673151.136-11.04
    -500.358.443840.008372923.21582134.889-20.61
    $ \beta $+504.54.514580.01573853.19932136.373-19.73
    +303.95.023580.01349563.18289150.54-11.4
    +153.455.382070.01181643.16882160.415-5.58
    -152.556.052270.008465613.13411178.6185.13
    -302.16.368190.006794073.11212187.06710.1
    -501.56.772880.004569663.07548197.75716.4
    $ \theta $+500.095.787540.01038013.07074171.681.05
    +300.0785.762980.01028773.10254170.9320.61
    +150.0695.743890.01021523.12725170.3510.61
    -150.0515.703820.01006113.17908169.133-0.45
    -300.0425.682760.0099793.20629168.493-0.83
    -500.035.653530.009864113, 24406167.606-1.35
    $ s $ +50451.600430.01623244.0189870.9432-58.24
    +30393.516820.01381683.69291129.89-23.55
    +1534.54.65880.01198963.43165154.538-9.04
    -1524.57.005390.0078352.78187179.1775.46
    -30217.838480.006334012.52404180.7476.38
    -50159.351260.003667252.02744176.3993.82
    $ \delta $+500.00755.768220.01011493.1489169.312-0.35
    +300.00655.750460.01012493.14046169.489-0.24
    +150.005755.737260.01013243.15161169.621-0.16
    -150.004255.71120.0101473.15388169.882-0.01
    -300.00355.698330.01015433.155170.0120.07
    -500.00255.681330.01016383.15647170.1830.17
    $ N $+501.056.04930.007759522.76778179.7055.77
    +300.915.937840.0085712.90253176.2823.75
    +150.8055.839390.009292353.01911173.2680.02
    -150.5955.586780.01115513.30832165.573-2.55
    -30.495.418930.01240133.49299160.49-5.54
    -500.355.123230.01460783.8047151.594-10.77
    $ \epsilon $+500.35.272470.01013323.15255159.446-6.15
    +300.265.438840.01013563.15261163.181-3.95
    +150.235.575080.01013763.15267166.293-2.12
    -150.175.890210.01014213.15284173.6642.21
    -300.146.079520.01014483.15295178.1994.88
    -500.16.387860.0101493.15313185.7439.32
    * Bold shows the most sensitive total inventory cost.
     | Show Table
    DownLoad: CSV
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